Use reference angles to find coordinates on the unit circle. Question: Find the coordinates of the point on the unit circle at an angle of θ. Give your answer in the form (x, y) and leave any fractions in fraction form. Provide your answer below: FEEDBACK MORE INSTRUCTIONS Content attribution
The Correct Answer and Explanation is:
(cos(θ), sin(θ))
Explanation:
The unit circle is a circle with a radius of 1 centered at the origin (0, 0) of the Cartesian coordinate plane. The coordinates of any point (x, y) on this circle can be determined by the angle, θ (theta), formed between the positive x axis and the line segment from the origin to that point. The angle is measured counterclockwise from the positive x axis.
By definition, the coordinates (x, y) for any angle θ on the unit circle are given by the trigonometric functions cosine and sine. Specifically, the x coordinate is equal to the cosine of the angle, and the y coordinate is equal to the sine of the angle. Therefore, for any given angle θ, the point on the unit circle is (cos(θ), sin(θ)).
To find the specific numerical values for these coordinates, we often use reference angles. A reference angle, denoted as θ’ (theta prime), is the acute angle that the terminal side of θ makes with the horizontal x axis. This method simplifies the process by relating any angle back to a familiar first quadrant angle (like 30°, 45°, or 60°).
The procedure is as follows:
- Determine the quadrant in which the angle θ lies.
- Calculate the reference angle θ’ by finding the smallest angle between the terminal side of θ and the x axis.
- Find the coordinates for the reference angle θ’, which will always be positive values from the first quadrant. For example, for a reference angle of 30°, the coordinates are (√3/2, 1/2).
- Apply the correct positive or negative signs to these coordinates based on the original angle’s quadrant. In Quadrant I, both x and y are positive. In Quadrant II, x is negative and y is positive. In Quadrant III, both are negative. In Quadrant IV, x is positive and y is negative.
This system allows us to use the values from a single quadrant to find coordinates for any angle on the entire unit circle.
